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Introduction

Practical Examples

MATLAB codes

SCIENTIFIC COMPUTING

FuSen Lin ()

Department of Computer Science & EngineeringNational Taiwan Ocean University

Scientific Computing, Fall 2007

FuSen Lin Scientific Computing

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Introduction

Practical Examples

MATLAB codes

Chapter 0: Introduction

Introduction to Scientific Computing

Three Practical Examples

Examples of MATLAB codes


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Introduction

Practical Examples

MATLAB codes

What is Scientific Computing?

To solve diversely mathematical problems which arise in

science and engineering.To design robust algorithms for finding approximate

solutions to the given problem with any powerful tools

(software).

Practical problems = Mathematical models= Numerical algorithms = (Reasonably approximate)solutions


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The Objective of Learning Scientific Computing?

To learn the potentialities of modern computational

methods for solving numerical problems arisen in science

and engineering.

To give students an opportunity to hone their skills in

programming and problem solving.

To help students understand the important subject of

errorsthat inevitably accompany scientific computing and

to arm them with methods for detecting, predicting, and

controlling these errors.To familiarize students with the intelligent use of powerful

software systems such as MATLAB, Maple, and

Mathematica.


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Introduction

Practical Examples

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The Goal of Scientific Computation

Numerical computations Pursue

Efficiency: the computational speed of a problem.Liability: the accuracy of computational results (within thegiven error tolerance).Stability: the stability of an algorithm and its applicablerange.

if use different algorithms for a problem, we may have

different accuracy, different speed, different applicable

range.

If we use the same algorithm with different procedures(coding) then the running time and accuracy may not be

the same.

We shall study and pursue the best algorithms.


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Practical Examples

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Problems in Scientific Computing

Finding-root problemsInterpolation problems

The method of least squares

Numerical Differentiation and Integration

Spline Approximation

Methods of Solving Linear Systems

Matrix Computations

Numerical solutions to ODE(Initial-Value Problems and

Boundary-Value Problems)

Numerical solutions to PDE (Eulers mehod, Multistepmethods, Rung-Kutta methods, Finite Difference Methods,

or Finite Element Method)

Optimization problems

Other computational problems


I t d ti E l 1

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Introduction

Practical Examples

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Example 1

Example 2

Example 3

Three Practical Examples

The finding-root Problem

The electrical circuit problem

The Least Squares Problem


Introduction Example 1

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Introduction

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Example 1

Example 2

Example 3

A finding-root problem

ExampleAn electric power cable is suspended (at points of equal

hight) from two towers that are 100 meters apart. The cable

is allowed to dip 10 meters in the middle. How long is the

cable? (It is known that the curve assumed by a suspended

cable is a Catenary()).

Proof.

Let x be the distance between each tower and the center of two

towers. When the y-axis passes through the lowest point of thecable, the curve can be expressed as

y = coshx

.

Here is a parameter to be determined. (continuing ).FuSen Lin Scientific Computing


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Introduction

Practical Examples

MATLAB codes

Example 1

Example 2

Example 3

A finding-root problem (continuing )

Proof.

(Continuing ) The conditions of this problem are

y(50) = y(0) + 10. The problem becomes to solve the equation

cosh

50

= + 10

for . Applying the methods of locating roots, we can obtain

= 126.632.

From the equation of catenary, the length of cable (arc length)

is easily computed by the method of calculus (numerical

integration) to be 102.619 meters (see MATLAB DEMO).



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Example 1

Example 2

Example 3

The Electrical Circuit Problem

ExampleA simple electrical network contains a number of resistances and a

single source of electromotive force (a battery) as shown in the

following figure.



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Example 1

Example 2

Example 3

The Electrical Circuit Problem

Using Kirchhoffs laws and Ohms law, we can write a system

of equations that govern this circuit. If x1, x2, x3, and x4 are theloop currents as shown, then the equations are

15x1 2x2 6x3 = 3002x1 + 12x2 4x3 x4 = 06x

1 4x

2+ 19x

3 9x

4= 0

x2 9x3 + 21x4 = 0

Systems of equations like this, even those that contain hundredunknowns, can be solved by using the methods of solving linear

systems.

6 2 2 40 4 2 2

0 0 2 50 0 0 3

x1x2x3x4

=

16693

The solution is x1 = 26.5, x2 = 9.35, x3 = 13.3, x4 = 6.13.



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Example 1

Example 2

Example 3

The Least Squares Problem

ExampleA physical experiment: Surface tension S in a liquid is known to

be a linear function of temperature T. For a particular liquid,

measurements have been made of the surface tension at

certain temperatures. The results were

T 0 10 20 30 40 80 90 95

S 68.0 67.1 66.4 65.6 64.6 61.8 61.0 60.0

How can the most probable values of the constants in the

equation S = aT + b be determined?

Methods for solving such problems are called the least

squares methods.


Introduction

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Solution of Example 1

% solution of example 1

syms x %

f = inline(x*cosh(50/x)-x-10);

lambda = fzero(f,100) % lambda = 126.6324

r=fzero(x.*cosh(50./x)-x-10,130) % finding root

syms a x %

a = 126.6324*cosh(x/126.6324);

s = (1+diff(a).2).0.5; % set up the arc length function

I = vpa(int(s,-50,50)) % find the length by integration


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Solution of Example 3

% Solution of Example3 for the least squares problem

% An example of Script Files

xdata = [0, 10, 20, 30, 40, 80, 90, 95]; %

ydata = [68.0, 67.1, 66.4, 65.6, 64.6, 61.8, 61.0, 60.0];

plot(xdata, ydata, o) % plot the datals = polyfit(xdata, ydata, 1); % call polyfit.m function

a = ls(1); b = ls(2);

d = a*xdata + b - ydata;

phi = sum(d.2);

x = 0:5:100;

y = polyval(ls, x); % compute the values of yhold on

plot(x, y,-r) % plot the curve of y

disp([a, b, phi] = )

[a, b, phi]


Introduction

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Function Files

function y = logsrs1(x, n)

% Date: 3/12/2001, Name: Fusen F. Lin

% This function computes log(2) by the series,

% log(1+x)= x-x2/2+x3/3-x4/4+x5/5-..., for n terms.

% Input : real x and integer n. Notice that:% the input x=1 to get log(2).

% Output: the desired value log(1+x).

tn = x; % The first term.

sn = tn; % The n-th partial sum.

for k = 1:1:n-1, % Use for-loop to sum the series.tn = -tn*x*k/(k+1); % compute it recursively.

s n = s n + t n ;

end

y = sn; % Output the final sum.


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Some Web Sites

The MathWorks, Inc. (MATLAB):http://www.mathworks.com.

Massachusetts Institute of Technology:

Department of EECS: http://www.eecs.mit.edu/grad/

MIT Open Courses: http://ocw.mit.edu/index.htmlChinese Web Sites of Open Courses:

http://www.myoops.org/twocw/mit/index.htm

Department of Mathematics, MIT:

http://math.mit.edu/research/index.html

The Computing Laboratory of Oxford University:

http://web.comlab.ox.ac.uk/oucl/

Department of Computer Science, Cornell University:

http://www.cs.cornell.edu/cv/


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